^{1}, E. A. Moss

^{1,a)}and B. W. Skews

^{1}

### Abstract

The aim of this work was to extend a previous investigation of the flow between two parallel disks (one of which was stationary) that have been subjected to a constant energy impact arising from a falling mass onto the upper disk assembly. Whereas the previous work considered the measurement of centreline pressures and distance between the plates only, for a single case, the current work in addition entailed monitoring of pressures at 45% and 90% of disk radius, under 28 combinations of drop height (100 to 1000 mm), drop mass (10 to 55 kg), and initial disk separation (3 to 10 mm), each with 5 repeat tests. Over the duration of the phenomenon (about 3.5 to 10 ms), four basic features were identified: (1) during initial impact under the dominance of temporal inertia, a preliminary pressure spike with peak pressures occurring at a displacement change of less than 0.25 mm from the initial disk separation; (2) an intermediate region with lower pressures; (3) pressure changes arising from a succession of elastic momentum exchanges (bounces) between the colliding masses; and (4) the final largest pressure spike towards the end of the phenomenon, where viscous effects dominate. Regions (1) and (4) became merged for smaller values of initial disk separation, with region (2) being obscured. A previously developed quasi-steady linear (QSL) model conformed satisfactorily with pressures measured at the centre of the lower disk; however, substantial deviations from radially parabolic pressure distributions were encountered over a range of operating parameters during the preliminary pressure phenomenon, unexpected because they implicitly conflict with the generally accepted concept of parallel flows and radially self-similar velocity profiles in such systems. Measurements of maximum pressures encountered during the preliminary and final pressure events agreed satisfactorily, both with the QSL model and with a simple but effective scaling analysis.

I. INTRODUCTION

II. ANALYSIS

III. EXPERIMENTATION

A. Test facility

B. Method

C. Main data processing

IV. RESULTS AND DISCUSSION

A. Gross momentum exchanges

B. Deviations from parabolic pressure distributions

C. Correlations between the QSL model and measurement for varying drop height

V. CONCLUSIONS

### Key Topics

- Viscosity
- 19.0
- Pressure measurement
- 11.0
- Acceleration measurement
- 9.0
- Kinematics
- 9.0
- Viscosity measurements
- 6.0

## Figures

Geometry of parallel disk arrangement.

Geometry of parallel disk arrangement.

(a) Components of test cell. (b) Test cell assembly. (c) Location of pressure transducers on lower disk. (a) and (b) reprinted with permission from E. A. Moss, A. Krassnokustki, B. W. Skews, and R. T. Paton, J. Fluid Mech.671, 384 (Year: 2011)10.1017/S0022112010005756. Copyright 2011, Cambridge University Press.

(a) Components of test cell. (b) Test cell assembly. (c) Location of pressure transducers on lower disk. (a) and (b) reprinted with permission from E. A. Moss, A. Krassnokustki, B. W. Skews, and R. T. Paton, J. Fluid Mech.671, 384 (Year: 2011)10.1017/S0022112010005756. Copyright 2011, Cambridge University Press.

Variations with drop height H of the ideal velocity of the dropped mass U ideal , calculated from drop height; the corresponding effective velocity U effective established on the basis of the energy dissipated; and the efficiency η, that is the ratio of energy dissipated to potential energy prior to the mass being dropped, for h 0 = 10 mm and M 1 = 30 kg.

Variations with drop height H of the ideal velocity of the dropped mass U ideal , calculated from drop height; the corresponding effective velocity U effective established on the basis of the energy dissipated; and the efficiency η, that is the ratio of energy dissipated to potential energy prior to the mass being dropped, for h 0 = 10 mm and M 1 = 30 kg.

Experimentally derived quantities corresponding to the reference test (H = 200 mm, h 0 ≃ 10 mm, M 1 = 30 kg): (a) variations of pressures (p t1, p t2, p t3a to p t3c ) and disk separation (h) with time (t) (thinner lines adjacent to the p t1 trace are uncertainty bounds); (b) variations of pressure measured at the centre of the lower disk (p t1), disk velocity ( ), and disk acceleration ( ), with time (t).

Experimentally derived quantities corresponding to the reference test (H = 200 mm, h 0 ≃ 10 mm, M 1 = 30 kg): (a) variations of pressures (p t1, p t2, p t3a to p t3c ) and disk separation (h) with time (t) (thinner lines adjacent to the p t1 trace are uncertainty bounds); (b) variations of pressure measured at the centre of the lower disk (p t1), disk velocity ( ), and disk acceleration ( ), with time (t).

(a) Variations of pressures with time t at the centre of the lower disk, showing comparisons between measurements (p t1) and predictions (p QSL, p v, p i) for the reference test (H = 200 mm, h 0 ≃ 10 mm, M 1 = 30 kg). (b) As for (a), but emphasizing the preliminary pressure spike (1).

(a) Variations of pressures with time t at the centre of the lower disk, showing comparisons between measurements (p t1) and predictions (p QSL, p v, p i) for the reference test (H = 200 mm, h 0 ≃ 10 mm, M 1 = 30 kg). (b) As for (a), but emphasizing the preliminary pressure spike (1).

Measured variations of pressure force (F p), reaction force of dropped mass onto upper disk assembly (F mass), and force accelerating the lower mass (F 2) with (a) time t and (b) disk separation (h) for H = 200 mm, h 0 ≃ 10 mm, and M 1 = 30 kg.

Measured variations of pressure force (F p), reaction force of dropped mass onto upper disk assembly (F mass), and force accelerating the lower mass (F 2) with (a) time t and (b) disk separation (h) for H = 200 mm, h 0 ≃ 10 mm, and M 1 = 30 kg.

Measured variations of pressure force (F p), reaction force of dropped mass onto upper disk assembly (F mass) and force accelerating the lower mass (F 2) with (a) time t and (b) disk separation (h) for H = 700 mm, h 0 ≃ 10 mm, and M 1 = 30 kg.

Measured variations of pressure force (F p), reaction force of dropped mass onto upper disk assembly (F mass) and force accelerating the lower mass (F 2) with (a) time t and (b) disk separation (h) for H = 700 mm, h 0 ≃ 10 mm, and M 1 = 30 kg.

Measured variations of pressure force (F p), reaction force of dropped mass onto upper disk assembly (F mass) and force accelerating the lower mass (F 2) with (a) time t and (b) disk separation (h) for H = 500 mm, h 0 ≃ 10 mm, and M 1 = 50 kg.

Measured variations of pressure force (F p), reaction force of dropped mass onto upper disk assembly (F mass) and force accelerating the lower mass (F 2) with (a) time t and (b) disk separation (h) for H = 500 mm, h 0 ≃ 10 mm, and M 1 = 50 kg.

Measured variations of pressure force (F p), reaction force of dropped mass onto upper disk assembly (F mass) and force accelerating the lower mass (F 2) with (a) time t and (b) disk separation (h) for H = 500 mm, h 0 ≃ 4 mm, and M 1 = 30 kg.

Measured variations of pressure force (F p), reaction force of dropped mass onto upper disk assembly (F mass) and force accelerating the lower mass (F 2) with (a) time t and (b) disk separation (h) for H = 500 mm, h 0 ≃ 4 mm, and M 1 = 30 kg.

Variations of measured dimensionless pressures (p t2/p t1, p t3ave/p t1) with radial position r/R for the main and preliminary pressure spikes, respectively, showing a parabolic pressure distribution for comparative purposes.

Variations of measured dimensionless pressures (p t2/p t1, p t3ave/p t1) with radial position r/R for the main and preliminary pressure spikes, respectively, showing a parabolic pressure distribution for comparative purposes.

(a) Variations of pressure p with disk separation h and radial position r for H = 200 mm, h 0 ≃ 10 mm, and M 1 = 30 kg. (b) As for (a), but embracing a range of h that emphasizes the preliminary pressure spike.

(a) Variations of pressure p with disk separation h and radial position r for H = 200 mm, h 0 ≃ 10 mm, and M 1 = 30 kg. (b) As for (a), but embracing a range of h that emphasizes the preliminary pressure spike.

(a) Variations of pressure p with disk separation h and radial position r for H = 700 mm, h 0 ≃ 10 mm, and M 1 = 30 kg. (b) As for (a), but embracing a range of h that emphasizes the preliminary pressure spike.

(a) Variations of pressure p with disk separation h and radial position r for H = 700 mm, h 0 ≃ 10 mm, and M 1 = 30 kg. (b) As for (a), but embracing a range of h that emphasizes the preliminary pressure spike.

(a) Variations of pressure p with disk separation h and radial position r for H = 500 mm, h 0 ≃ 10 mm, and M 1 = 50 kg. (b) As for (a), but embracing a range of h that emphasizes the preliminary pressure spike.

(a) Variations of pressure p with disk separation h and radial position r for H = 500 mm, h 0 ≃ 10 mm, and M 1 = 50 kg. (b) As for (a), but embracing a range of h that emphasizes the preliminary pressure spike.

(a) Variations of pressure p with disk separation h and radial position r for H = 500 mm, h 0 ≃ 4 mm, and M 1 = 30 kg. (b) As for (a), but embracing a range of h that emphasizes the preliminary pressure spike.

(a) Variations of pressure p with disk separation h and radial position r for H = 500 mm, h 0 ≃ 4 mm, and M 1 = 30 kg. (b) As for (a), but embracing a range of h that emphasizes the preliminary pressure spike.

Variations of (a) measured (p t1) and (b) predicted (p QSL) pressures at the centre of the lower disk, with drop height H, and disk separation h, for h 0 ≃ 10 mm and M 1 = 30 kg.

Variations of (a) measured (p t1) and (b) predicted (p QSL) pressures at the centre of the lower disk, with drop height H, and disk separation h, for h 0 ≃ 10 mm and M 1 = 30 kg.

Variations of (a) measured (p t1) and (b) predicted (p QSL) pressures at the centre of the lower disk, with drop height H, and disk separation h, for h 0 ≃ 10 mm and M 1 = 30 kg, emphasizing the preliminary pressure spike.

Variations of (a) measured (p t1) and (b) predicted (p QSL) pressures at the centre of the lower disk, with drop height H, and disk separation h, for h 0 ≃ 10 mm and M 1 = 30 kg, emphasizing the preliminary pressure spike.

Variations of measured and predicted maximum pressures at the centre of the lower disk (p t1max and p QSLmax) with initial disk separation h 0 during: (a) the major and (b) preliminary pressure spikes, respectively, for M 1 = 30 kg and H = 500 mm. The solid lines are power law least square regression fits to the sets of data.

Variations of measured and predicted maximum pressures at the centre of the lower disk (p t1max and p QSLmax) with initial disk separation h 0 during: (a) the major and (b) preliminary pressure spikes, respectively, for M 1 = 30 kg and H = 500 mm. The solid lines are power law least square regression fits to the sets of data.

Variations of measured and predicted maximum pressures at the centre of the lower disk (p t1max and p QSLmax) with impact mass M 1 during (a) the major and (b) preliminary pressure spikes, respectively, for h 0 ≃ 10 mm and H = 500 mm. The solid lines are power law least square regression fits to the sets of data.

Variations of measured and predicted maximum pressures at the centre of the lower disk (p t1max and p QSLmax) with impact mass M 1 during (a) the major and (b) preliminary pressure spikes, respectively, for h 0 ≃ 10 mm and H = 500 mm. The solid lines are power law least square regression fits to the sets of data.

Variations of measured and predicted maximum pressures at the centre of the lower disk (p t1max and p QSLmax) with drop height H during: (a) the major and (b) preliminary pressure spikes, respectively, for M 1 = 30 kg and h 0 ≃ 10 mm. The solid lines are power law least square regression fits to the sets of data.

Variations of measured and predicted maximum pressures at the centre of the lower disk (p t1max and p QSLmax) with drop height H during: (a) the major and (b) preliminary pressure spikes, respectively, for M 1 = 30 kg and h 0 ≃ 10 mm. The solid lines are power law least square regression fits to the sets of data.

## Tables

Range of parameters covered during testing, with the bold numbers representing the values at which two of the parameters were fixed while varying the third.

Range of parameters covered during testing, with the bold numbers representing the values at which two of the parameters were fixed while varying the third.

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